Problem: Find the value of $k$ so that the line $3x + 5y + k = 0$ is tangent to the parabola $y^2 = 24x.$
Solution: Solving for $x$ in $3x + 5y + k = 0,$ we get
\[x = -\frac{5y + k}{3}.\]Substituting into $y^2 = 24x,$ we get
\[y^2 = -40y - 8k,\]or $y^2 + 40y + 8k = 0.$  Since we have a tangent, this quadratic will have a double root, meaning that its discriminant will be 0.  This give us $40^2 - 4(8k) = 0,$ so $k = \boxed{50}.$